RUI: Skeins on Surfaces

本研究主要涉及几何拓扑学和量子物理学的交叉学科领域。许多令人振奋的猜想来自物理学，它们的数学解将会引起理论物理学家的兴趣。该项目的第二部分涉及DNA和蛋白质等生物聚合物的拓扑特征。从制药的角度来看，这一点很重要，因为一些药物可以针对影响特定生物功能的拓扑特征进行设计。除了研究目标外，该项目还有很强的教育成分。许多提出的问题都是针对本科生的研究。通过她的研究、教学和其他外展活动，她打算扩大数学的影响范围，例如，扩展到历史上代表性不足的群体和其他通常不接触尖端数学的受众。这个项目将探索考夫曼斜线代数及其推广在多大程度上可以作为量子拓扑和双曲几何之间的中介。PI将学习Kauffman skein代数的表示理论，特别是来自Witten-Reshetikhin-Turaev理论的表示。长期的首要目标是构造和分类Kauffman Skein代数的所有表示，这是本项目将推进的目标。该项目考虑了考夫曼斜线代数及其推广的代数结构(例如，允许曲面上有圆弧的那些)。此外，该项目还包括研究在DNA和蛋白质等生物聚合物中可能存在哪些类型的拓扑复杂结构，如结、链接和非平面图。

On the main, this research project lies in the broad interdisciplinary area between geometric topology and quantum physics. Many of the motivating conjectures come from physics, and their mathematical solutions would be of interest to theoretical physicists. A second part of the project concerns the topological characteristics of biopolymers like DNA and proteins. This can be important from a pharmaceutical perspective, as some drugs can be designed to target topological characteristics which affect specific biological functions. Besides its research goals, the project has a strong educational component. Many of the proposed problems are intended for research with undergraduate students. Through her research, teaching and other outreach activities, the PI intends to expand the reach of mathematics, for example to historically under-represented groups and to other audiences not usually exposed to cutting edge mathematics.This project will explore the extent to which the Kauffman skein algebra and its generalizations can serve as intermediaries between quantum topology and hyperbolic geometry. The PI will study the representation theory of the Kauffman skein algebra, paying particular attention to the representation coming from the Witten-Reshetikhin-Turaev theory. The long-term, overarching goal is to construct and classify all representations of the Kauffman skein algebra, a goal which this project will advance. The project considers the algebraic structure of the Kauffman skein algebra and of its generalizations (e.g., ones that allow arcs on the surface). In addition, the project includes problems investigating which types of topologically complex structures, like knots, links, and non-planar graphs, are possible in biopolymers like DNA and proteins.